*2.1. Solving Fluid Flow Using LBM*

In this work, as mentioned earlier, LBM is used to obtain the flow field due to natural convection. In LBM, the macroscopic variables such as fluid pressure and the velocity, are computed from the fluid-PDF, *fn*(**<sup>x</sup>**,*<sup>t</sup>*), which are evaluated by solving the Boltzmann kinetic equation on a discrete lattice mesh. Here, *fn*(**<sup>x</sup>**,*<sup>t</sup>*) is the probability of finding a fluid particle at a lattice position, **x**, and at a time, *t*, moving with a discrete velocity, **c***n* (the subscript *n* indicates the PDF number), which is selected in such a way that after time step Δ*t*, the particle arrives at the *nth* neighboring grid point [21]. The single-relaxation-time lattice Boltzmann equation (LBE) with external body force term is given by [37]

$$f\_{\rm ll}(\mathbf{x} + \mathbf{c}\_{\rm tr}\Delta t, t + \Delta t) = f\_{\rm tr}(\mathbf{x}, t) - \frac{1}{\lambda} \Big( f\_{\rm tr}(\mathbf{x}, t) - f\_{\rm tr}^{\rm cl}(\mathbf{x}, t) \Big) + \frac{w\_{\rm tr}\Delta t}{c\_{\rm s}^2} \Big( \Big( \mathbf{f}^{\rm th}(\mathbf{x}, t) + \mathbf{f}^{\rm cl}(\mathbf{x}, t) \Big) \cdot \mathbf{c}\_{\rm tr} \Big) \tag{1}$$

where λ is the relaxation time, *feqn* (**<sup>x</sup>**,*<sup>t</sup>*) is the equilibrium distribution functions, *wn* is the weighing function, and *cs* is the sound speed. **<sup>f</sup>**th(**<sup>x</sup>**,*<sup>t</sup>*) and **<sup>f</sup>**fl(**<sup>x</sup>**,*<sup>t</sup>*) in Equation (1) represent the buoyancy force and the hydrodynamic force (due to the no-slip BC on the hexagonal surfaces) source terms, respectively. Through the Chapman–Enskog analysis, the above equation recovers the Navier–Stokes equations in the low Mach number limit, |**u**|/*cs* 1 [14]. The relation between the fluid kinematic viscosity, ν, and relaxation time, λ, is given by

$$
\nu = c\_s^2 \Delta t \left(\lambda - \frac{1}{2}\right). \tag{2}
$$

To model the buoyancy force, **<sup>f</sup>**th(**<sup>x</sup>**, *t*), the Boussinesq approximation is used as follows

$$\mathbf{f^{th}} = \rho\_0 \boldsymbol{\beta} (T - T\_0) \mathbf{g^{\uparrow}} \tag{3}$$

where ρ0 is the initial value for fluid density, *T*0 is the initial fluid temperature, β is the fluid thermal expansion coefficient at *T*0, *T* is the fluid temperature field, and **^ j** is the unit vector in the vertical direction (*y* − direction). The hydrodynamic force term, **<sup>f</sup>**fl(**<sup>x</sup>**, *t*) of Equation (1) is obtained with SPM. After solving for *fn*(**<sup>x</sup>**, *t*), the fluid density and the velocity fields, ρ(**<sup>x</sup>**, *t*) and **<sup>u</sup>**(**<sup>x</sup>**, *t*), are obtained from

$$\rho(\mathbf{x},t) = \sum\_{n=0}^{b} f\_{n\prime} \quad \mathbf{u}(\mathbf{x},t) = \frac{1}{\rho} \sum\_{n=0}^{b} f\_n \mathbf{c}\_n. \tag{4}$$
